×

Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians. (English) Zbl 0896.14025

Summary: In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians \(G(k,n)\) to some Gorenstein toric Fano varieties \(P(k,n)\) with conifold singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections \(X \subset G(k,n)\) of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational quantum cohomology of Grassmannians.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M10 Complete intersections
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Avram, A. C.; Candelas, P.; Jancic, D.; Mandelberg, M., On the Connectedness of the Moduli Space of Calabi-Yau Manifolds, Nucl. Phys. B, 465, 458 (1996), (hep-th/9511230) · Zbl 0896.14027
[2] A.C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke, The web of Calabi-Yau hypersurfaces in toric varieties, hep-th/9703003; A.C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke, The web of Calabi-Yau hypersurfaces in toric varieties, hep-th/9703003 · Zbl 0896.14026
[3] Batyrev, V. V., On classifications of smooth projective toric varieties, Tôhoku Math. J., 43, 569 (1991) · Zbl 0792.14026
[4] Batyrev, V. V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom., 3, 493 (1994) · Zbl 0829.14023
[5] Batyrev, V. V.; Borisov, L. A., On Calabi-Yau Complete Intersections in Toric Varieties, (Andreatta, M.; Peternell, T., Higher Dimensional Complex Geometry (1996)), 37, (alg-geom/9412017) · Zbl 0908.14015
[6] Batyrev, V. V.; Borisov, L. A., Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, (Greene, B.; Yau, S-T., Mirror Symmetry II (1997), AMS and Int. Press), 71 · Zbl 0927.14019
[7] Batyrev, V. V.; van Straten, D., Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Commun. Math. Phys., 168, 493 (1995), (alg-geom/9307010) · Zbl 0843.14016
[8] Batyrev, V. V.; Ciocan-Fontanine, I.; Kim, B.; van Straten, D., Mirror symmetry and toric degenerations of partial flag manifolds (1997), preprint · Zbl 1022.14014
[9] A. Bertram, Quantum Schubert calculus, to appear in Adv. Math.; A. Bertram, Quantum Schubert calculus, to appear in Adv. Math.
[10] Behrend, K., Gromov-Witten invariants in algebraic geometry, Invent. Math., 127, 601 (1997) · Zbl 0909.14007
[11] L.A. Borisov, Towards mirror symmetry of Calabi-Yau complete intersections in Gorenstein toric Fano varieties, alg-geom/9310001.; L.A. Borisov, Towards mirror symmetry of Calabi-Yau complete intersections in Gorenstein toric Fano varieties, alg-geom/9310001.
[12] I. Brunner, M. Lynker and R. Schimmrigk, Dualities and phase transitions for Calabi-Yau 3-folds and 4-folds, hep-th/9703182.; I. Brunner, M. Lynker and R. Schimmrigk, Dualities and phase transitions for Calabi-Yau 3-folds and 4-folds, hep-th/9703182. · Zbl 0925.32011
[13] T.-M. Chiang, B.R. Greene, M. Gross and Y. Kanter, Black hole condensation and the web of Calabi-Yau manifolds, hep-th/9511204.; T.-M. Chiang, B.R. Greene, M. Gross and Y. Kanter, Black hole condensation and the web of Calabi-Yau manifolds, hep-th/9511204.
[14] Candelas, P.; Dale, A.; Lütken, C. A.; Schimmrigk, R., Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B, 298, 493 (1988)
[15] Candelas, P.; Green, P.; Hübsch, T., Rolling among Calabi-Yau vacua, Nucl. Phys. B, 330, 49 (1990) · Zbl 0985.32502
[16] Clemens, H., Double solids, Adv. Math., 47, 107 (1983) · Zbl 0509.14045
[17] T. Eguchi, K. Hori and C.-S. Xiong, Gravitational quantum cohomology, hep-th/9605225, to appear in Int. J. Mod. Phys. A.; T. Eguchi, K. Hori and C.-S. Xiong, Gravitational quantum cohomology, hep-th/9605225, to appear in Int. J. Mod. Phys. A.
[18] W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology, preprint, alg-geom/9608011.; W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology, preprint, alg-geom/9608011. · Zbl 0898.14018
[19] Givental, A., Equivariant Gromov-Witten invariants, IMRN, No. 13, 613 (1996), (alg-geom/9603021) · Zbl 0881.55006
[20] A. Givental, Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture, alg-geom/9612001.; A. Givental, Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture, alg-geom/9612001.
[21] A. Givental, A mirror theorem for toric complete intersections, alg-geom/9701016.; A. Givental, A mirror theorem for toric complete intersections, alg-geom/9701016.
[22] Lakshmibai, V., Degeneration of flag varieties to toric varieties, C.R. Acad. Sci. Paris, 321, 1229 (1995) · Zbl 0858.14026
[23] Gonciulea, N.; Lakshmibai, V., Degenerations of flag and Schubert varieties to toric varieties (1996), preprint · Zbl 0909.14028
[24] Gonciulea, N.; Laksmibai, V., Schubert varieties, toric varieties, and ladder determinantal varieties (1996), preprint · Zbl 0909.14028
[25] Greene, B. R.; Morrison, D. R.; Strominger, A., Black hole condensation and the unification of string vacua, Nucl. Phys. B, 451, 109 (1995), hep-th/9504145 · Zbl 0908.53041
[26] D. Morrison, Through the looking glass, alg-geom/9705028.; D. Morrison, Through the looking glass, alg-geom/9705028.
[27] Kim, B., Quantum hyperplane section theorem (1997), preprint
[28] Li, J.; Tian, G., Virtual moduli cycles and Gromov-Witten invariants (1996), preprint
[29] Libgober, A.; Teitelbaum, J., Lines on Calabi-Yau Complete Intersections, mirror symmetry and Picard-Fuchs equations, Duke Math. J., 69, 29 (1993), Int. Math. Res. Notes · Zbl 0789.14005
[30] Lynker, M.; Schimmrigk, R., Conifold transitions and mirror symmetries, Nucl. Phys. B, 484, 562 (1997), (hep-th/9511058) · Zbl 0925.14013
[31] Reid, M., The moduli space of 3-folds with \(K = 0\) may nevertheless be irreducible, Math. Ann., 278, 329 (1987) · Zbl 0649.14021
[32] van Straten, D., A quintic hypersurface in \(P^4\) wioth 130 nodes, Topology, 32, 857 (1993) · Zbl 0801.14015
[33] Strominger, A., Massless black hole condensation and the unification of string vacua, Nucl. Phys. B, 451, 96 (1995), hep-th/9504090 · Zbl 0908.53041
[34] Sturmfels, B., Gröbner Bases and Convex Polytopes, (Univ. Lect. Notes, Vol. 8 (1996), AMS) · Zbl 0856.13020
[35] Tjøtta, E., Rational curves on the space of determinantal nets of conics, Thesis (1997), Bergen
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.